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localized coinvariant

Let BB be a kk-bialgebra and EE a right BB-comodule algebra with coaction ρ:E→E⊗B\rho:E\to E\otimes B. Let j:E→Eμj : E\to E_\mu be a biflat affine localization in the sense that it is a homomorphism of rings (affiness), and that the extension of scalars along jj is an exact localization functor whose right adjoint is also exact. If BB is a Hopf algebra of coordinate functions on an alegbraic kk-group where kk-is a field, and EE is the coordinate algebra of an affine kk-variety with regular GG-action, then we can not restrict the action to an arbitrary open sets, but only to the GG-invariant open sets. In our case, this would mean that the coaction does not necessarily extend to the localization EμE_\mu respecting the algebra structure.

For this reason, one introduces the class of ρ\rho-compatible localizations: a localization of a BB-comodule algebra (E,ρ)(E,\rho) is coaction compatible localization if the coaction extends to an algebra morphism ρμ:Eμ→Eμ⊗B\rho_\mu:E_\mu\to E_\mu\otimes B such that the diagram

commmutes. If such an extension exists it is unique and EμE_\mu is automatically a coaction.

The algebra of localized coinvariants is the subalgebra EμcoB⊂EμE_\mu^{co B}\subset E_\mu of ρμ\rho_\mu-coinvariants in EμE_\mu. It is important to notice that taking coinvariants and localization do not commute: there is no localization EcoB→EμcoBE^{co B}\to E^{co B}_\mu in general. A BB-comodule algebra which is ρ\rho-compatible may have many new BB-coinvariants which do not come in any sense as images under some sort of a localization from the BB-coinvariants of EE. This localization can be used to get more local information on the noncommutative quotient space for coactions of Hopf algebra. In some cases one can construct a cover of the quotient spaces by affine charts whose coordinate algebras are the algebras of localized coinvariant for varying compatible localizations of the original comodule algebra; this can be further generalized by replacing the comodule algebra by a more general object which is also glued from the charts, but this requires consideration of the compatibility at the level of the localization functors and the actions are then globally expressed via an action of a monoidal category (which is still locally induced by a coaction).

The notion of compatibility of coactions and Ore localizations and the localized coinvariants have been introduced in 1997 when starting working on